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What is extreme finitism?

Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects.

Strict Finitism is a form of finitism that rejects objects known as ‘countably infinite’.

Extreme Finitism goes much further as it rejects all uses of the concept of ‘infinity’.

Extreme Finitism is the philosophy that only accepts the existence of finite objects, although some objects may be defined as having a variable/indeterminable size. Furthermore, it opposes the use of words that imply that ‘infinity’ is a valid concept.

These words include ‘infinitely many’, ‘an infinite number of’, ‘infinite’, ‘to infinity’, ‘continuous’, ‘analogue’, ‘forever’, ‘eternally’, ‘infinitesimal’ and all other words and phrases that imply ‘infinity’.

You might wonder how the counting numbers cannot be infinite when there is obviously no last number. But knowing how to count and knowing there is no last number does not make ‘infinity’ a reality.

We can write down instructions on how to count natural numbers starting from 1. Lots of different people can look at these instructions and start counting. The instructions might not specify when to stop counting and so there is no ‘last number’. However, only a finite number of people can ever follow these instructions and each can only count up to a finite number.

The set of instructions can be called an ‘algorithm’. Everything about this algorithm and its use is entirely finite.

We might represent this algorithm as the series 1 + 1 + 1 +… or as the sequence 1, 2, 3, 4, …

This demonstrates that having a counting algorithm with no end point specified does not imply that an actual infinity can/must somehow occur, despite what our intuition might tell us.